As others have mentioned, the Gengar from the mission are shiny locked, like with all static encounters, though Gengar is able to spawn beneath the tower in Wild Zone 20. However, I think it's worthwhile to mention the odds for this.
Serebii cites a 1 in 67 chance for each of the Alpha spawns to be a Gengar. If the player has obtained the Shiny Charm, wild Pokemon have a 1 in 1024 chance of being shiny. Putting this together, there is a 1 in 68,608 chance for a spawn to be a shiny Gengar, and with 2 spawns available, it's approximately a 1 in 34,304 chance to get a shiny Gengar with each reroll of the area. Not having the Shiny Charm makes your odds 4× worse.
If you left your Switch on and continuously rerolled the area nonstop, you would probably expect to get one after several days, depending on how fast your load time is—it takes me about 20 seconds to change the time at the nearby bench, which comes out to an expected total time of just over 9 real life days straight—but all that's disrupted because you'd most likely generate other shiny Pokemon before you find a shiny Gengar. This is incredibly inconvenient since the fact that shinies don't get rerolled forces you to frequently check for and remove any other shinies, adding an unpredictable but significant amount of extra time and preventing the hunt from being automated with any tools.
As a result... In theory, yes, you can find a wild shiny Gengar if you seriously cannot trade under any circumstances, but in practice, I really don't recommend trying to. I'm sympathetic that it's a hassle to buy online functionality and search for a trustworthy trade partner, but obtaining one in-game is much more cumbersome.
